ABSTRACT
We introduced the concept of a group as a mathematical means to describe symmetries; in fact, we learned to distinguish different types of symmetry in geometrical patterns by looking in each case at the underlying group of transformations leaving invariant the given pattern. However, when the notion of a group was shaped and formulated historically, this was not done to describe geometrical patterns – for obvious reasons: One does not need an abstract mathematical instrument to describe facts which are obvious to one’s bare eyes. On the other hand, if it comes to describing “hidden” symmetries such a mathematical tool might be very useful, and this was recognized for the first time in the theory of polynomials.† Mathematicians like Joseph-Louis de Lagrange (1736–1813), Niels Hendrik Abel (1802–1829) and, above all, Evariste Galois (1811–1832) were well aware of the fact that each polynomial f possesses an underlying hidden symmetry, namely, the symmetry between those roots of f which cannot be algebraically distinguished.
